An approach to the simultaneous measurement of refractive-index (RI) and temperature changes using optical ring resonators is proposed and theoretically demonstrated. With a liquid-core silica ring resonator as an example, two different-order whispering gallery modes (WGMs) might differ in not only RI but also temperature sensitivities. Thus, a second-order sensing matrix should be defined based on these WGMs to determine RI and temperature changes simultaneously. The analysis shows that the RI and temperature detection limits can be achieved on the order of 10^{-7} RI unit and 10^{-3} K at a wavelength of approximately 780 nm.